Measuring Strategy Alpha from a Zero-Value Starting Portfolio
Summary
The document examines why percentage-based alpha is undefined when a strategy begins with no cash and no assets. It describes a simple paper test in which borrowed funds buy assets and performance is assessed from cash plus the liquidation value of holdings. Comparing percentage changes with a benchmark then requires a nonzero initial denominator, creating an arbitrary choice of starting capital.
The question considers setting that capital using beta and a tolerated probability of the portfolio reaching zero, but notes that volatility alone does not capture the asymmetric tail risk relevant to ruin; distributional features such as the mean, skewness, and kurtosis may also matter. The reply observes that conventional brokers require collateral, so zero-capital starts are uncommon in traditional trading. It suggests treating signup time as an economic contribution for cases such as token giveaways. This is an informal framing rather than a defined alpha estimator or risk model, and it offers no empirical analysis or general method for selecting capital.
Key ideas
- Percentage returns and alpha cannot use a zero starting portfolio value as their denominator.
- Choosing an initial capital amount affects calculated percentage performance.
- Volatility alone may not describe the chance of a portfolio reaching zero.
- Broker collateral requirements usually make a literal zero-capital trading start unusual.
- The proposed value for signup time is an illustrative convention, not a standard performance measure.
Tags
Full text
# How to adjust a strategy's alpha assuming a zero-value starting portfolio (\$0 cash, \$0 assets)? # How to adjust a strategy's alpha assuming a zero-value starting portfolio (\$0 cash, \$0 assets)? A simple paper test of a trading strategy is to assume one borrows all money to purchase assets and see if trading increases the liquidation value of the portfolio (cash + liquidation value of assets). However, in order to calculate the trading strategy's alpha, one must take the percentage change compared to the benchmark's. Since one starts with $0, this creates a divide-by-zero situation. One can remedy this situation by assuming starting cash > \$0 and use that in the denominator, but it isn't obvious what number to choose for this starting cash. To exemplify the difficulty let's say one chooses a starting cash amount that is some function of the beta -- so as to reduce the risk that the strategy will be forced to borrow money during trading, i.e. that the total portfolio value will hit $0 liquidation value. This function would also take, as input, the level of acceptable risk that the strategy will be forced borrow money during trading. Moreover, since beta is the standard deviation, and one is attempting to avoid a $0 balance, it is inadequate input due to the symmetry of its deviation about the mean. There must be additional input to the function, e.g. mean, skewness, kurtosis, etc. There is probably some work on this in the literature but I've been unable to find it. ## Answer by Brian B (score 1) https://quant.stackexchange.com/a/68752 The $0 case has not historically come up, and has not tended to be relevant to "the literature". Prime brokers and retail brokers all required at least a certain amount of cash (or other collateral) to open accounts and start trading. These days, we have some crypto trading platforms that attract clientele with a giveaway of some (small) amount of cryptotokens just for signing up, which could legitimately be considered a $0 starting point. I might suggest instead avoiding the issue by assigning a nonzero value to the 5 minutes needed to sign up as a client, by applying 1/12 times the local hourly minimum wage.
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